diff --git a/doc/c-code.tex b/doc/c-code.tex index 7af3d6b2f..b29e1a9d2 100644 --- a/doc/c-code.tex +++ b/doc/c-code.tex @@ -196,6 +196,27 @@ \subsection{$\gamma$ matrices} \end{split} \] +\subsection{Pauli matrices} +\[ +\begin{split} + \tau^1 = + \begin{pmatrix} + 0 & 1 \\ + 1 & 0 \\ + \end{pmatrix},\quad + \tau^2 = + \begin{pmatrix} + 0 & -i \\ + i & 0 \\ + \end{pmatrix},\quad + \tau^3 = + \begin{pmatrix} + 1 & 0 \\ + 0 & -1 \\ + \end{pmatrix} +\end{split} +\] + \subsection{Flavour Split Doublet Operator} The convention we use internally for the flavour split doublet Dirac @@ -210,8 +231,20 @@ \subsection{Flavour Split Doublet Operator} \] The relation between the two is given by \[ -D_h' = (1+i\tau^2)D_h(1-i\tau^2)\, . +D_h' = \frac{1}{\sqrt{2}}(1+i\tau^2)\ D_h\ \frac{1}{\sqrt{2}}(1-i\tau^2)\, . +\] +The implementation is then such that first the source $\xi$ is multiplied +with +\[ +\xi\to\xi=\frac{1}{\sqrt{2}}(1-i\tau^2)\xi +\] +on which $D_h$ is inverted and the solution $\phi$ is obtained. The +solution is then multiplied +\[ +\phi \to \phi=\frac{1}{\sqrt{2}}(1+i\tau^2)\phi \] +to obtain the result for $D_h'$. + The convention $D_h'$ is the one of \cite{Chiarappa:2006ae} with $\bar\mu = \mu_\sigma$ and $\bar\epsilon = \mu_\delta$. {\ttfamily invert\_doublet} inverts with the source first set for the upper @@ -221,6 +254,71 @@ \subsection{Flavour Split Doublet Operator} lower, upper, lower flavour. The former two correspond to the source in the upper flavour, the latter to the source in the lower flavour. +\subsection{Stochastic Volume Sources} + +In order to compute disconnected contributions volume (all spin, +colour, space and time) sources are implemented. In this case only one +inversion is required. The volume sources are generated with gaussian +noise ($\sigma=1$) in real and imaginary part of the whole source +spinor. Note that the normalisation with $1/\sqrt{2}$ is \emph{not} +done and needs to be taken care off in the analysis. + +For the hopping parameter noise reduction method the following is +needed: Following the notation in Ref.~\cite{Boucaud:2008xu} the +operator can be written as +\[ +D_h' = A + H = (1+H\cdot B)\cdot A,\qquad B=1/A\, . +\] +where $H^\dagger = \gamma_5 H \gamma_5$ and +\[ +A = 1 + i\gamma_5\tau^1\tilde\mu_\sigma + \tau^3\tilde\mu_\delta +\] +where in the hopping parameter representation $\tilde\mu_\sigma = +2\kappa\mu_\sigma$ and $\tilde\mu_\delta=2\kappa\mu_\delta$. $A$ can +be inverted easily to +\[ +A^{-1}=\frac{1-i\gamma_5 \tau^1 \tilde\mu_\sigma - \tau_3 + \tilde\mu_\delta}{1+\tilde\mu_\sigma^2-\tilde\mu_\delta^2} +\] +It follows that +\[ +1/D_h' = B-BHB+B(HB)^2-B(HB)^3+1/D_h'(HB)^4\ . +\] +Then, since $\gamma_5$ commutes with $B$ one can evaluate the last +term stochastically for any $\gamma$ and or colour matrix $X$ like +\[ +X (1/D_h')(HB)^4 = \lim_{R\to\infty}\left[(\gamma_5 (B^{\dagger} H )^4 + \gamma_5 \xi)^* X \phi\right]_R +\] +where +\[ +\phi = (D_h')^{-1}\xi +\] +and +\[ +B(\tilde\mu_\sigma)^\dagger \equiv B(-\tilde\mu_\sigma)\ . +\] +The remaining terms can be computed exactly. For any source $\xi$ we +therefore have to generate +\[ +\xi_r = \gamma_5 (B^{\dagger} H )^4\gamma_5 \xi. +\] + + +%(ii)-------------------------------------------- +%Use source in all flavour spin colour space time (Z2xZ2) +% +%Let u,v be flavour indices (c,s) equivalent +% +%M \phi = \xi with flavour explicit M_{uv) \phi_v = \xi_u +% +% Then required quantity is +% Tr (X M^{-1} )_{uv} where X is diagonal in flavour +% and Tr is sum over colour, spin, space (at a given time) +% = Tr( \xi^*_v X \ph_u ) +% = Tr ( {(g_5 (B^{\dag} H)^4 g_5 \xi)_v}^* X \phi_u) +% for uv element + %%% Local Variables: